© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n116 K11n118

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6

Gauss Code:

{1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 5, -9, 4, -10, 7, -11, 8}

DT (Dowker-Thistlethwaite) Code:

4 10 -14 -18 -16 2 -20 -22 -8 -12 -6

Alexander Polynomial:

- 3t-2 + 9t-1 - 11 + 9t - 3t2

Conway Polynomial:

1 - 3z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{1020, 10162, ...}

Determinant and Signature:

{35, 2}

Jones Polynomial:

q-3 - 2q-2 + 4q-1 - 5 + 6q - 6q2 + 5q3 - 4q4 + 2q5

Other knots (up to mirrors) with the same Jones Polynomial:

{10138, ...}

A2 (sl(3)) Invariant:

q-10 + 2q-4 + 1 - 2q4 - 2q8 + q10 + 2q16

HOMFLY-PT Polynomial:

2a-4 + 2a-4z2 - 3a-2 - 5a-2z2 - 2a-2z4 + 1 - z2 - z4 + a2 + a2z2

Kauffman Polynomial:

2a-6z2 - a-5z + 2a-5z3 + a-5z5 + 2a-4 - 7a-4z2 + 10a-4z4 - 4a-4z6 + a-4z8 + a-3z - 7a-3z3 + 9a-3z5 - 4a-3z7 + a-3z9 + 3a-2 - 14a-2z2 + 17a-2z4 - 11a-2z6 + 3a-2z8 + 2a-1z - 4a-1z3 + a-1z5 - 2a-1z7 + a-1z9 + 1 - z2 + 3z4 - 6z6 + 2z8 + 5az3 - 7az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-3, -2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                 2 j = 9               2   j = 7             3 2   j = 5           3 2     j = 3         3 3       j = 1       3 4         j = -1     1 2           j = -3   1 3             j = -5   1               j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 117]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 117]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 19, 8, 18], > X[9, 17, 10, 16], X[2, 11, 3, 12], X[13, 21, 14, 20], X[15, 22, 16, 1], > X[17, 9, 18, 8], X[19, 13, 20, 12], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 117]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 5, -9, 4, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 117]]
Out[5]=	DTCode[4, 10, -14, -18, -16, 2, -20, -22, -8, -12, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 117]][t]
Out[6]=	3 9 2 -11 - -- + - + 9 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 117]][z]
Out[7]=	2 4 1 - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 117]], KnotSignature[Knot[11, NonAlternating, 117]]}
Out[9]=	{35, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 117]][q]
Out[10]=	-3 2 4 2 3 4 5 -5 + q - -- + - + 6 q - 6 q + 5 q - 4 q + 2 q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 138], Knot[11, NonAlternating, 117]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 117]][q]
Out[12]=	-10 2 4 8 10 16 1 + q + -- - 2 q - 2 q + q + 2 q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 117]][a, z]
Out[13]=	2 2 4 2 3 2 2 2 z 5 z 2 2 4 2 z 1 + -- - -- + a - z + ---- - ---- + a z - z - ---- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 117]][a, z]
Out[14]=	2 2 2 3 2 3 2 z z 2 z 2 2 z 7 z 14 z 2 2 2 z 1 + -- + -- - a - -- + -- + --- - z + ---- - ---- - ----- + 4 a z + ---- - 4 2 5 3 a 6 4 2 5 a a a a a a a a 3 3 4 4 5 5 5 7 z 4 z 3 4 10 z 17 z 2 4 z 9 z z > ---- - ---- + 5 a z + 3 z + ----- + ----- - 4 a z + -- + ---- + -- - 3 a 4 2 5 3 a a a a a a 6 6 7 7 8 5 6 4 z 11 z 2 6 4 z 2 z 7 8 z > 7 a z - 6 z - ---- - ----- + a z - ---- - ---- + 2 a z + 2 z + -- + 4 2 3 a 4 a a a a 8 9 9 3 z z z > ---- + -- + -- 2 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 117]], Vassiliev[3][Knot[11, NonAlternating, 117]]}
Out[15]=	{-3, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 117]][q, t]
Out[16]=	3 1 1 1 3 1 2 3 q 3 4 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 3 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 11 4 > 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n117

K11n116 K11n118